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The objective of the present book of essays is to convey to the intelligent nonmathematician something of the nature, development, and use of mathe matical concepts, particularly those that have found application in current scientific research. The idea of assembling such a volume goes back at least to 1974, when it was discussed by the then-newly-formed Joint Projects Committee for Mathematics (JPCM) of the American Mathematical Soci ety, the Mathematical Association of America, and the Society for Indus trial and Applied Mathematics. Currently, the nine members of the JPCM are Saunders Mac Lane (Chairman) of the University of Chicago, Frederick J. Almgren, Jr. of Princeton University, Richard D. Anderson of Louisiana State University, George E. Carrier of Harvard University, Hirsh G. Cohen ofthe International Business Machines Corporation, Richard C. DiPrima of Rensselaer Polytechnic Institute, Robion C. Kirby of the University of California at Berkeley, William H. Kruskal of the University of Chicago, and George D. Mostow of Yale University. The JPCM decided to make production of this volume its first major proj ect and requested the Conference Board of the Mathematical Sciences (CBMS), of which its three sponsoring societies are all member organiza tions, to approach the National Science Foundation on its behalffor support of the undertaking. A proposal submitted by the C BMS in December 1974 and in revised form in July 1975 was granted by the Foundation in May 1976, and work on assembling the volume got under way.
Designed for a junior-senior level course for mathematics majors, including those who plan to teach in secondary school. The first chapter presents several finite geometries in an axiomatic framework, while Chapter 2 continues the synthetic approach in introducing both Euclids and ideas of non-Euclidean geometry. There follows a new introduction to symmetry and hands-on explorations of isometries that precedes an extensive analytic treatment of similarities and affinities. Chapter 4 presents plane projective geometry both synthetically and analytically, and the new Chapter 5 uses a descriptive and exploratory approach to introduce chaos theory and fractal geometry, stressing the self-similarity of fractals and their generation by transformations from Chapter 3. Throughout, each chapter includes a list of suggested resources for applications or related topics in areas such as art and history, plus this second edition points to Web locations of author-developed guides for dynamic software explorations of the Poincaré model, isometries, projectivities, conics and fractals. Parallel versions are available for "Cabri Geometry" and "Geometers Sketchpad".
By presenting teacher profiles and sample lessons from across the country, this book shows that the NCTM standards reflect successful practices of teachers at the "grass roots".